We define the notion of stochastic stability, already present in the literature in the context of smooth dynamical systems, for invariant measures of cellular automata perturbed by a random noise, and the notion of strongly stochastically stable cellular automaton. We study these notions on basic examples (nilpotent cellular automata, spreading symbols) using different methods inspired by those presented in {MST19}. We then show that this notion of stability is not trivial by proving that a Turing machine cannot decide if a given invariant measure of a cellular automaton is stable under a uniform perturbation.
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Marsan et al. (2024) studied this question.
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